In Search of Truth (Hard Version)

The only difference between easy and hard versions is the maximum number of queries. In this version, you are allowed to ask at most $$$1000$$$ queries.

This is an interactive problem.

You are playing a game. The circle is divided into $$$n$$$ sectors, sectors are numbered from $$$1$$$ to $$$n$$$ in some order. You are in the adjacent room and do not know either the number of sectors or their numbers. There is also an arrow that initially points to some sector. Initially, the host tells you the number of the sector to which the arrow points. After that, you can ask the host to move the arrow $$$k$$$ sectors counterclockwise or clockwise at most $$$1000$$$ times. And each time you are told the number of the sector to which the arrow points.

Your task is to determine the integer $$$n$$$ — the number of sectors in at most $$$1000$$$ queries.

It is guaranteed that $$$1 \le n \le 10^6$$$.

After reading the description of the input, you may ask queries. Queries can be of two types:

1. "+ k" ($$$0 \le k \le 10^9$$$) — ask to move the arrow $$$k$$$ sectors clockwise.
2. "- k" ($$$0 \le k \le 10^9$$$) — ask to move the arrow $$$k$$$ sectors counterclockwise.

After each query, you should read an integer $$$x$$$ ($$$1 \le x \le n$$$) — the number of the current sector to which the arrow points.

You are allowed to make at most $$$1000$$$ queries in total.

If you make too many queries, you will get Wrong answer.

After printing a query or the answer, do not forget to output a the end of line and flush the output. Otherwise, you will get Idleness limit exceeded. To do this, use:

• fflush(stdout) or cout.flush() in C++;
• System.out.flush() in Java;
• flush(output) in Pascal;
• stdout.flush() in Python;
• see the documentation for other languages.